Mark variogram of spatio-temporal point processes in the analysis of fire hotspots in Itapuã do Oeste, Brazilian Amazon
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Abstract
Point processes are widely used statistical tools for analyzing spatial, temporal, or spatiotemporal patterns in several phenomena, such as crime occurrences, seismic events, plant species distribution, and fire hotspots. However, the analysis of these events requires distinguishing genuine spatiotemporal interaction structures from intensity trends in order to avoid misleading interpretations of clustering patterns. The objective of this study is to analyze the spatiotemporal behavior of fire hotspots in the Brazilian Amazon region, using the municipality of Itapuã do Oeste between 2018 and 2019 as a case study. Following an existing methodology, we apply the mark variogram, a technique traditionally used for marked point processes, to data from a spatiotemporal point process (STPP). The approach consists of decomposing the STPP into two marked point processes: one considering occurrence times as marks of spatial locations and the other considering spatial locations as marks of occurrence times, followed by fitting a Gaussian model. To improve the inferential interpretation of the method, we extend this approach by constructing simulation envelopes under both homogeneous (HPP) and non-homogeneous Poisson process (NHPP) null models. All analyses were performed in the R environment. The results revealed a spatial dependence structure for fire hotspots up to approximately 4 km. In the temporal domain, however, the observed clustering pattern was mainly associated with first-order intensity variation rather than true second-order dependence. Under the HPP null model, the empirical mark variogram lay significantly outside the simulation envelopes, suggesting apparent structural clustering. However, under the NHPP null model, the empirical function remained entirely within the envelope limits, indicating that the observed clustering behavior can be explained by non-homogeneous intensity patterns. These results demonstrate that incorporating non-homogeneous simulation envelopes into the analysis of spatiotemporal mark variograms improves inferential reliability by reducing false clustering interpretations. Consequently, the proposed approach provides a more robust framework for understanding fire dynamics and may support environmental monitoring and management strategies in critical ecosystems such as the Amazon rainforest.
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